Multiple GCDs. probabilistic analysis of the plain algorithm

Abstract

This paper provides a probabilistic analysis of an algorithm which computes the gcd of inputs (with ≥ 2), with a succession of − 1 phases, each of them being the Euclid algorithm on two entries. This algorithm is both basic and natural, and two kinds of inputs are studied: polynomials over the finite field Fq and integers. The analysis exhibits the precise probabilistic behaviour of the main parameters, namely the number of iterations in each phase and the evolution of the length of the current gcd along the execution. We first provide an average-case analysis. Then we make it even more precise by a distributional analysis. Our results rigorously exhibit two phenomena: (i) there is a strong difference between the first phase, where most of the computations are done and the remaining phases; (ii) there is a strong similarity between the polynomial and integer cases.

Probabilistic analyses of the plain multiple gcd algorithm

“…We then conclude the paper with Section 10. This paper is an extended version of a previous short paper (Berthé et al, 2013) which appeared in the Proceedings of the ISSAC'2013 Conference. For the polynomial case, we provide here two proofs which do not appear in the short version (namely the proofs of Proposition 11 and Theorem 5), and we develop thoroughly the analysis in the integer case: the average-case analysis was only briefly described in the short version, and the distributional analysis provided here is completely new.…”